Hermann Schlichting • Klaus Gersten

Boundary-LayerTheoryWith contributionsfrom Egon Krause and Herbert Oertel Jr.

Translated by Katherine Mayes

8th Revised and Enlarged EditionWith 287 Figures and 22 Tables

Springer

Contents

Introduction XIX

Part I. Fundamentals of Viscous Flows

1. Some Features of Viscous Flows 31.1 Real and Ideal Fluids 31.2 Viscosity 41.3 Reynolds Number 61.4 Laminar and Turbulent Flows 121.5 Asymptotic Behaviour at Large Reynolds Numbers 141.6 Comparison of Measurements

Using the Inviscid Limiting Solution 141.7 Summary 26

2. Fundamentals of Boundary—Layer Theory 292.1 Boundary-Layer Concept 292.2 Laminar Boundary Layer on a Flat Plate at Zero Incidence . . 302.3 Turbulent Boundary Layer on a Flat Plate at Zero Incidence. 332.4 Fully Developed Turbulent Flow in a Pipe 362.5 Boundary Layer on an Airfoil 382.6 Separation,of the Boundary Layer 392.7 Overview of the Following Material 48

3. Field Equations for Flows of Newtonian Fluids 513.1 Description of Flow Fields 513.2 Continuity Equation 523.3 Momentum Equation 523.4 General Stress State of Deformable Bodies 533.5 General State of Deformation of Flowing Fluids 573.6 Relation Between Stresses and Rate of Deformation 623.7 Stokes Hypothesis 653.8 Bulk Viscosity and Thermodynamic Pressure 663.9 Navier-Stokes Equations 68

X Contents

3.10 Energy Equation 693.11 Equations of Motion

for Arbitrary Coordinate Systems (Summary) 733.12 Equations of Motion

for Cartesian Coordinates in Index Notation 763.13 Equations of Motion in Different Coordinate Systems 79

4. General Properties of the Equations of Motion 834.1 Similarity Laws 834.2 Similarity Laws for Flow with Buoyancy Forces

(Mixed Forced and Natural Convection) 864.3 Similarity Laws for Natural Convection 904.4 Vorticity Transport Equation 914.5 Limit of Very Small Reynolds Numbers 934.6 Limit of Very Large Reynolds Numbers 944.7 Mathematical Example of the Limit Re —> oo 964.8 Non-Uniqueness of Solutions of the Navier-Stokes Equations. 99

5. Exact Solutions of the Navier—Stokes Equations 1015.1 Steady Plane Flows 101

5.1.1 Couette-Poiseuille Flows 1015.1.2 Jeffery-Hamel Flows

(Fully Developed Nozzle and Diffuser Flows) 1045.1.3 Plane Stagnation-Point Flow 1105.1.4 Flow Past a Parabolic Body 1155.1.5 Flow Past a Circular Cylinder 115

5.2 Steady Axisymmetric Flows 1165.2.1 Circular Pipe Flow (Hagen-Poiseuille Flow) 1165.2.2 Flow Between Two Concentric Rotating Cylinders . . . . 1175.2.3 Axisymmetric Stagnation-Point Flow 1185.2.4 Flow at a Rotating Disk 1195.2.5 Axisymmetric Free Jet '. 124

5.3 Unsteady Plane Flows 1265.3.1 Flow at a Wall Suddenly Set into Motion

/(First Stokes Problem) 1265.3.2 Flow at an Oscillating Wall

(Second Stokes Problem) 1295.3.3 Start-up of Couette Flow 1305.3.4 Unsteady Asymptotic Suction 1315.3.5 Unsteady Plane Stagnation-Point Flow 1315.3.6 Oscillating Channel Flow 137

5.4 Unsteady Axisymmetric Flows 1395.4.1 Vortex Decay 1395.4.2 Unsteady Pipe Flow 139

5.5 Summary 141

Contents XI

Part II. Laminar Boundary Layers

6. Boundary—Layer Equations in Plane Flow;Plate Boundary Layer 1456.1 Setting up the Boundary-Layer Equations 1456.2 Wall Friction, Separation and Displacement 1506.3 Dimensional Representation

of the Boundary-Layer Equations 1526.4 Friction Drag 1556.5 Plate Boundary Layer 156

7. General Properties and Exact Solutionsof the Boundary—Layer Equationsfor Plane Flows 1657.1 Compatibility Condition at the Wall 1667.2 Similar Solutions of the Boundary-Layer Equations 167

7.2.1 Derivation of the Ordinary Differential Equation 167A Boundary Layers with Outer Flow 169B Boundary Layers Without Outer Flow 172

7.2.2 Wedge Flows 1727.2.3 Flow in a Convergent Channel 1747.2.4 Mixing Layer 1757.2.5 Moving Plate 1777.2.6 Free Jet 1777.2.7 Wall Jet 180

7.3 Coordinate Transformation 1827.3.1 Gortler Transformation 1827.3.2 v. Mises Transformation 1837.3.3 Crocco Transformation 184

7.4 Series Expansion of the Solutions 1847.4.1 Blasius Series 1847.4.2 Gortler Series 186

7.5 Asymptotic Behaviour of Solutions Downstream 1877.5.1 /Wake Behind Bodies ' 1877.5.2 Boundary Layer at a Moving Wall 190

7.6 Integral Relations of the Boundary Layer 1917.6.1 Momentum—Integral Equation 1917.6.2 Energy-Integral Equation 1927.6.3 Moment-of-Momentum Integral Equations 194

8. Approximate Methods for Solving the Boundary—LayerEquations for Steady Plane Flows 1958.1 Integral Methods 196

XII Contents

8.2 Stratford's Separation Criterion 2028.3 Comparison of the Approximate Solutions

with Exact Solutions 2028.3.1 Retarded Stagnation-Point Flow 2028.3.2 Divergent Channel (Diffuser) 2048.3.3 Circular Cylinder Flow 2058.3.4 Symmetric Flow past a Joukowsky Airfoil 207

9. Thermal Boundary LayersWithout Coupling of the Velocity Fieldto the Temperature Field 2099.1 Boundary—Layer Equations for the Temperature Field 2099.2 Forced Convection for Constant Properties 2119.3 Effect of the Prandtl Number 2159.4 Similar Solutions of the Thermal Boundary Layer 2189.5 Integral Methods for Computing the Heat Transfer 2239.6 Effect of Dissipation;

Distribution of the Adiabatic Wall Temperature 226

10. Thermal Boundary Layerswith Coupling of the Velocity Fieldto the Temperature Field 23110.1 Remark : 23110.2 Boundary-Layer Equations 23110.3 Boundary Layers with Moderate Wall Heat Transfer

(Without Gravitational Effects) 23310.3.1 Perturbation Calculation 23310.3.2 Property Ratio Method (Temperature Ratio Method) . 23710.3.3 Reference Temperature Method 240

10.4 Compressible Boundary Layers(Without Gravitational Effects) 24110.4.1 Physical Property Relations . 24110.4.2 Simple Solutions of the Energy Equation 24410.4.3 Transformations of the Boundary-Layer Equations . . . 24610.4.4 Similar Solutions 24910.4.5 Integral Methods 25810.4.6 Boundary Layers in Hypersonic Flows 263

10.5 Natural Convection 26510.5.1 Boundary-Layer Equations 26510.5.2 Transformation of the Boundary-Layer Equations . . . . 27010.5.3 Limit of Large Prandtl Numbers (Tw = const) 27110.5.4 Similar Solutions 27310.5.5 General Solutions 27710.5.6 Variable Physical Properties 27810.5.7 Effect of Dissipation 280

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Contents XIII

10.6 Indirect Natural Convection 28110.7 Mixed Convection 284

11. Boundary-Layer Control (Suction/Blowing) 29111.1 Different Kinds of Boundary-Layer Control 29111.2 Continuous Suction and Blowing 295

11.2.1 Fundamentals 29511.2.2 Massive Suction 29711.2.3 Massive Blowing 29911.2.4 Similar Solutions 30211.2.5 General Solutions 307

1. Plate Flow with Uniform Suction or Blowing 3072. Airfoil ; 309

11.2.6 Natural Convection with Blowing and Suction 31011.3 Binary Boundary Layers 311

11.3.1 Overview 31111.3.2 Basic Equations 31211.3.3 Analogy Between Heat and Mass Transfer 31611.3.4 Similar Solutions 317

12. Axisymmetric and Three-Dimensional Boundary Layers . . 32112.1 Axisymmetric Boundary Layers 321

12.1.1 Boundary-Layer Equations 32112.1.2 Mangier Transformation 32312.1.3 Boundary Layers

on Non-Rotating Bodies of Revolution 32412.1.4 Boundary Layers on Rotating Bodies of Revolution . . . 32712.1.5 Free Jets and Wakes 331

12.2 Three-Dimensional Boundary Layers 33512.2.1 Boundary-Layer Equations 33512.2.2 Boundary Layer at a Cylinder 34112.2.3 Boundary Layer at a Yawing Cylinder 34212.2A Three-Dimensional Stagnation Point 34412.2.5 Boundary Layers in Symmetry Planes 34512.2.6 General Configurations 345

13. Unsteady Boundary Layers 34913.1 Fundamentals 349

13.1.1 Remark 34913.1.2 Boundary-Layer Equations 35013.1.3 Similar and Semi-Similar Solutions 35113.1.4 Solutions for Small Times (High Frequencies) 35213.1.5 Separation of Unsteady Boundary Layers 35313.1.6 Integral Relations and Integral Methods 354

13.2 Unsteady Motion of Bodies in a Fluid at Rest 355

XIV Contents

13.2.1 Start-Up Processes 35513.2.2 Oscillation of Bodies in a Fluid at Rest 362

13.3 Unsteady Boundary Layers in a Steady Basic Flow 36513.3.1 Periodic Outer Flow 36513.3.2 Steady Flow with a Weak Periodic Perturbation 36713.3.3 Transition Between Two Slightly Different Steady

Boundary Layers 36913.4 Compressible Unsteady Boundary Layers 370

13.4.1 Remark 37013.4.2 Boundary Layer Behind a Moving

Normal Shock Wave 37113.4.3 Flat Plate at Zero Incidence with Variable Free Stream

Velocity and Wall Temperature 373

14. Extensions to the Prandtl Boundary-Layer Theory 37714.1 Remark 37714.2 Higher Order Boundary-Layer Theory 37914.3 Hypersonic Interaction 38914.4 Triple-Deck Theory 39214.5 Marginal Separation 40314.6 Massive Separation 408

Part III. Laminar—Turbulent Transition

15. Onset of Turbulence (Stability Theory) 41515.1 Some Experimental Results

on the Laminar-Turbulent Transition 41515.1.1 Transition in the,Pipe Flow 41515.1.2 Transition in the Boundary Layer 419

15.2 Fundamentals of Stability Theory 42415.2.1 Remark -..'. 42415.2.2 Fundamentals of Primary Stability Theory 42515.2.3 Orr-Sommerfeld Equation 42715.2.4 Curve of Neutral Stability

and the Indifference Reynolds Number 434a Plate Boundary Layer '.'... 436b Effect of Pressure Gradient 445c Effect of Suction 457d Effect of Wall Heat Transfer 460e Effect of Compressibility 463f Effect of Wall Roughness 467g Further Effects 472

15.3 Instability of the Boundary Layer• for Three-Dimensional Perturbations 473

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Contents XV

15.3.1 Remark 47315.3.2 Fundamentals of Secondary Stability Theory 47615.3.3 Boundary Layers at Curved Walls 47915.3.4 Boundary Layer at a Rotating Disk 48315.3.5 Three-Dimensional Boundary Layers 485

15.4 Local Perturbations 491

Part IV. Turbulent Boundary Layers

16. Fundamentals of Turbulent Flows 49716.1 Remark 49716.2 Mean Motion and Fluctuations 49916.3 Basic Equations for the Mean Motion of Turbulent Flows . . . . 502

16.3.1 Continuity Equation 50216.3.2 Momentum Equations (Reynolds Equations) 50316.3.3 Equation for the Kinetic Energy

of the Turbulent Fluctuations (fc-Equation) 50516.3.4 Thermal Energy Equation 508

16.4 Closure Problem 50916.5 Description of the Turbulent Fluctuations 510

16.5.1 Correlations 51016.5.2 Spectra and Eddies 51116.5.3 Turbulence of the Outer Flow 51316.5.4 Edges of Turbulent Regions and Intermittence 514

16.6 Boundary-Layer Equations for Plane Flows 515

17. Internal Flows 51917.1 Couette Flow 519

17.1.1 Two-Layer Structure of the Velocity Fieldand the Logarithmic Overlap Law 519

17.1.2 Universal Laws of the Wall 52417.1.3 Friction Law 53617.1.4 Turbulence Models 53817.1.5 Heat Transfer 541

17.2 Fully Developed Internal Flows (A = const) 54317.2.1 Channel Flow '..... 54317.2.2 Couette-Poiseuille Flows 54417.2.3 Pipe Flow 549

17.3 Slender-Channel Theory 554

18. Turbulent Boundary LayersWithout Coupling of the Velocity Fieldto the Temperature Field 55718.1 Turbulence Models 557

XVI Contents

18.1.1 Remark 55718.1.2 Algebraic Turbulence Models 55918.1.3 Turbulent Energy Equation 56018.1.4 Two-Equation Models 56218.1.5 Reynolds Stress Models 56518.1.6 Heat Transfer Models 56818.1.7 Low-Reynolds-Number Models 57018.1.8 Large-Eddy Simulation

and Direct Numerical Simulation 57118.2 Attached Boundary Layers 572

18.2.1 Layered Structure 57218.2.2 Boundary-Layer Equations Using

the Defect Formulation 57418.2.3 Friction Law and Characterisitic Quantities

of the Boundary Layer 57718.2.4 Equilibrium Boundary Layers 58018.2.5 Boundary Layer on a Plate at Zero Incidence 582

18.3 Boundary Layers with Separation 58918.3.1 Stratford Flow 58918.3.2 Quasi-Equilibrium Boundary Layers 591

18.4 Computation of Boundary Layers Using Integral Methods . . . 59418.4.1 Direct Method 59418.4.2 Inverse Method 597

18.5 Computation of Boundary Layers Using Field Methods 59818.5.1 Attached Boundary Layers 59818.5.2 Boundary Layers with Separation 60118.5.3 Low-Reynolds-Number Turbulence Models 60218.5.4 Additional Effects 604

18.6 Computation of Thermal Boundary Layers 60618.6.1 Fundamentals 60618.6.2 Computation of Thermal Boundary Layers

Using Field Methods " 609

19. Turbulent Boundary Layerswith Coupling of the Velocity Fieldto the Temperature Field 61119.1 Fundamental Equations 611

19.1.1 Time Averaging for Variable Density 61119.1.2 Boundary-Layer Equations 613

19.2 Compressible Turbulent Boundary Layers 61719.2.1 Temperature Field 61719.2.2 Overlap Law 61919.2.3 Skin-Friction Coefficient and Nusselt Number 62119.2.4 Integral Methods for Adiabatic Walls 623

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Contents XVII

19.2.5 Field Methods 62519.2.6 Shock-Boundary-Layer Interaction 625

19.3 Natural Convection 627

20. Axisymmetric and Three—DimensionalTurbulent Boundary Layers 63120.1 Axisymmetric Boundary Layers 631

20.1.1 Boundary-Layer Equations 63120.1.2 Boundary Layers Without Body Rotation 63220.1.3 Boundary Layers with Body Rotation 635

20.2 Three-Dimensional Boundary Layers 63720.2.1 Boundary-Layer Equations 63720.2.2 Computation Methods 64120.2.3 Examples 643

21. Unsteady Turbulent Boundary Layers 64521.1 Averaging and Boundary-Layer Equations 64521.2 Computation Methods 64821.3 Examples 649

22. Turbulent Free Shear Flows 65322.1 Remark 65322.2 Equations for Plane Free Shear Layers 65522.3 Plane Free Jet 659

22.3.1 Global Balances 65922.3.2 Far Field 66022.3.3 Near Field 66522.3.4 Wall Effects 665

22.4 Mixing Layer 66722.5 Plane Wake , 66922.6 Axisymmetric Free Shear Flows 671

22.6.1 Basic Equations 67122.6.2 Free Jet 67222.6.3 Wake 674

22.7 Buoyant Jets 67522.7.1 Plane Buoyant Jet 67522.7.2 Axisymmetric Buoyant Jet : . . . . 677

22.8 Plane Wall Jet 677

XVIII Contents

Part V. Numerical Methods in Boundary—Layer Theory

23. Numerical Integration of the Boundary—Layer Equations . 68323.1 Laminar Boundary Layers 683

23.1.1 Remark 68323.1.2 Note on Boundary-Layer Transformations 68423.1.3 Explicit and Implicit Discretisation 68523.1.4 Solution of the Implicit Difference Equations 68923.1.5 Integration of the Continuity Equation 69123.1.6 Boundary-Layer Edge and Wall Shear Stress 69123.1.7 Integration of the Transformed Boundary-Layer

Equations Using the Box Scheme 69223.2 Turbulent Boundary Layers 695

23.2.1 Method of Wall Functions 69523.2.2 Low-Reynolds-Number Turbulence Models 700

23.3 Unsteady Boundary Layers 70123.4 Steady Three-Dimensional Boundary Layers 703

List of Frequently Used Symbols 709

References 717

Index 795